How pointed can magnetized ferrofluid tips be?

The tip radius $R_{\mathrm{tip}}$ of ferrofluids deformed by magnetization is typically ∼100 μm, much larger than in conducting electrified liquids, even though in both cases the capillary stress $2\gamma /R_{\mathrm{tip}}$ balances the electrostatic or magnetostatic stress. The reason for the relative dullness of magnetic tips appears to be the saturation of magnetization, $M_{\max }=M_{\mathrm{sat}}$. This hypothesis lacks quantitative experimental confirmation but is supported theoretically by the recently proposed high field asymptote $R_{\mathrm{tip}}\to \frac{4\gamma }{{\mu }_o{M}_{\mathrm{sat}}^2}$ obtained for isolated drops in strong and uniform magnetic fields. Here we confirm experimentally the controlling role of the magnetic saturation effect under more general conditions, by showing that the dimensionless group ${\mu }_o{M}_{\mathrm{sat}}^2{R}_{\mathrm{tip}}/\left(4\gamma \right)$ is always a quantity of order unity, even for ferrofluid menisci subject to the highly nonuniform magnetic fields created by supporting them at the tip of a ferrous needle facing a strong magnet. Four different ferrofluids are used, with high, medium, and low surface tensions, including highly polar and nonpolar solvents. Substantial changes in $M_{\mathrm{sat}}$ are achieved by letting the suspending solvent evaporate, with record tip radii down to 9.5 μm demonstrated at high volume fractions of magnetic particles.

Figure 1

Simulated magnitude of the magnetic field around a ferrous needle (OD 360 µm, ID 180 µm) axially located 6 mm away from the face of the magnet. Notice the saddle point stabilizing the meniscus.

Figure 2

Axial dependence of the magnetic flux density at the axis: in the absence of the ferrous needle (black line) and including the needle (color lines correspond to needle tip separations of 4, 5, 6, 7, 8, and 10 mm from the face of the magnet). The inset shows the local magnetic field near the face of the steel needle with position directed from the face of the steel needle.

Figure 3

Four magnetostatic unevaporated menisci pinned to a blunt steel needle. From left to right, the solvents are glycerol, water, dodecylbenzene, and $\mathrm{EMIm}\text{−}\mathrm{NT}{\mathrm{f}}_2$. Note the characteristic onion-dome shape in (c).

Figure 4

Magnetization as a function of the applied field for four ferrofluids.

Figure 5

Frames extracted at 10-s intervals from footage of a fast evaporating water droplet. The tip sharpens as the droplet dries, and the volume fraction of magnetic particles rises. Eventually the volume fraction of particles is so high that the meniscus begins to solidify, preferentially nucleating at the tip and base of the meniscus.

Figure 6

Collected results from three experiments with evaporating menisci. Top (a)–(c): quickly evaporating aqueous ferrofluid. Middle (d)–(f): aqueous ferrofluid drying in a humidified environment. Bottom (g)–(i): slowly evaporating (original) EFH1 droplet. The continuous lines in the right images compare the asymptotic prediction (2) with the measured $R_{\mathrm{tip}}$, showing in all cases that their ratio is of order unity.

Figure 7

An overview of the measurement of curvature, with the sharpest tip found for EFH1. From the raw image (a) we extract the contour coordinates, represented in (b) as discrete white points extending only 81 μm from the base of the meniscus. A parabola is fit through those points, shown in a lighter tone (red in the web edition) in (c). The quality of the adjustment between the discretized contour data and the parabolic fit is confirmed in the linear relation shown in (d) between the square root of the axial variable and the radial variable $\left(R^2=0.9991\right)$.

Figure 8

Magnetic model of the driest aqueous ferrofluid droplet in the rapidly evaporating experiment. Magnetic flux density is magnified within the droplet as the permeability is significantly higher than in air.

Figure 9

Two DDBFF menisci stabilized on a magnetized razor's edge. The sole change between the high-density (top) and low-density (bottom) images is the volume deposited in the magnetic sink.